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Average behavior of Fourier coefficients of cusp forms

Author: Guangshi Lü
Journal: Proc. Amer. Math. Soc. 137 (2009), 1961-1969
MSC (2000): Primary 11F30, 11F11, 11F66
Published electronically: December 30, 2008
MathSciNet review: 2480277
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ a_0(n)$ and $ b_0(n)$ be the normalized Fourier coefficients of the two holomorphic Hecke eigenforms $ f(z)\in S_{2k}(\Gamma)$ and $ \varphi(z)\in S_{2l}(\Gamma)$ respectively. In 1999, Fomenko studied the following average sums of $ a_0(n)$ and $ b_0(n)$:

$\displaystyle \sum_{n \leq x}a_0(n)^3, \quad \sum_{n \leq x}a_0(n)^2b_0(n), \quad \sum_{n \leq x}a_0(n)^2b_0(n)^2, \quad \sum_{n \leq x}a_0(n)^4.$      

In this paper, we are able to improve on Fomenko's results.

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  • 1. Daniel Bump and David Ginzburg, Symmetric square 𝐿-functions on 𝐺𝐿(𝑟), Ann. of Math. (2) 136 (1992), no. 1, 137–205. MR 1173928, 10.2307/2946548
  • 2. K. Chandrasekharan and Raghavan Narasimhan, Functional equations with multiple gamma factors and the average order of arithmetical functions, Ann. of Math. (2) 76 (1962), 93–136. MR 0140491
  • 3. Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 0340258
  • 4. O. M. Fomenko, Fourier coefficients of parabolic forms, and automorphic 𝐿-functions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 237 (1997), no. Anal. Teor. Chisel i Teor. Funkts. 14, 194–226, 231 (Russian, with Russian summary); English transl., J. Math. Sci. (New York) 95 (1999), no. 3, 2295–2316. MR 1691291, 10.1007/BF02172473
  • 5. Stephen Gelbart and Hervé Jacquet, A relation between automorphic representations of 𝐺𝐿(2) and 𝐺𝐿(3), Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471–542. MR 533066
  • 6. Dorian Goldfeld, Automorphic forms and 𝐿-functions for the group 𝐺𝐿(𝑛,𝐑), Cambridge Studies in Advanced Mathematics, vol. 99, Cambridge University Press, Cambridge, 2006. With an appendix by Kevin A. Broughan. MR 2254662
  • 7. H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367–464. MR 701565, 10.2307/2374264
  • 8. Hervé Jacquet and Joseph Shalika, Rankin-Selberg convolutions: Archimedean theory, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989) Israel Math. Conf. Proc., vol. 2, Weizmann, Jerusalem, 1990, pp. 125–207. MR 1159102
  • 9. Henryk Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 1474964
  • 10. Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214
  • 11. Carlos J. Moreno and Freydoon Shahidi, The fourth moment of Ramanujan 𝜏-function, Math. Ann. 266 (1983), no. 2, 233–239. MR 724740, 10.1007/BF01458445
  • 12. R.A. Rankin, Contributions to the theory of Ramanujan's function $ \tau(n)$ and similar arithmetical functions, II. The order of the Fourier coefficients of the integral modular forms, Proc. Cambridge Phil. Soc., 35(1939), 357-372.
  • 13. Freydoon Shahidi, Third symmetric power 𝐿-functions for 𝐺𝐿(2), Compositio Math. 70 (1989), no. 3, 245–273. MR 1002045
  • 14. Goro Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3) 31 (1975), no. 1, 79–98. MR 0382176

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Additional Information

Guangshi Lü
Affiliation: Department of Mathematics, Shandong University, Jinan, Shandong, 250100, People’s Republic of China

Keywords: Fourier coefficients of cusp forms, Gelbart-Jacquet lift, $L$-function
Received by editor(s): May 30, 2008
Received by editor(s) in revised form: August 28, 2008
Published electronically: December 30, 2008
Additional Notes: This work was supported by the National Natural Science Foundation of China (Grant No. 10701048).
Communicated by: Ken Ono
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.